Temporal resolution of birth rate analysis in zooplankton and its implications for identifying strong interactions in ecology

Abstract Expanding on Haeckel's classical definition, ecology can be defined as the study of strong and weak interactions between the organism and the environment, hence the need for identifying strong interactions as major drivers of population and community dynamics. The solution to this problem is facilitated by the fact that the frequency distribution of interaction strengths is highly skewed, resulting in few or, according to Liebig's law of the minimum, just one strong interaction. However, a single strong interaction often remains elusive. One of the reasons may be that, due to the ever‐present dynamics of ecological systems, a single strong interaction is likely to exist only on relatively short time intervals, so methods with sufficient temporal resolution are required. In this paper, we study the temporal resolution of contribution analysis of birth rate in zooplankton, a method to assess the relative strength of bottom‐up (food) versus top‐down (predation) effects. Birth rate is estimated by the Edmondson–Paloheimo model. Our test system is a population of the cladoceran Bosmina longirostris inhabiting a small northern lake with few planktivorous predators, and thus likely controlled by food. We find that the method's temporal resolution in detecting bottom‐up effects corresponds well to the species' generation time, and the latter seems comparable to the lifetime of a single strong interaction. This enables one to capture a single strong interaction “on the fly,” right during its time of existence. We suggest that this feature, the temporal resolution of about the lifetime of a single strong interaction, may be a generally desirable property for any method, not only the one studied here, intended to identify and assess strong interactions. Success in disentangling strong interactions in ecological communities, and thus solving one of the key issues in ecology, may critically depend on the temporal resolution of the methods used.


| INTRODUC TI ON
Ecology has long been known as the science of interactions. This can be seen from the continuity of the two popular definitions separated by more than 100 years, namely that of Haeckel (1866: 286, quoted in Nikol'skii, 2014) that ecology is the science of the relationships between the organism and its organic and inorganic environment and that of Krebs (1972Krebs ( , 2014 that ecology is the "scientific study of the interactions that determine the distribution and abundance of organisms"; common to both is the notion of interactions (or relationships). An important point to be added from more recent work is that ecological interactions are not of the same strength; some of them are much stronger than the others, resulting in a substantial nonzero variance in the interaction strength (Allesina & Tang, 2015;May, 1972;Pimm, 2002). The strong interactions are a major immediate driver of population and community dynamics (Paine, 1992). The weak interactions are no less important because, collectively, they play a primary role in the stability of ecological communities (de Ruiter et al., 1995;Downing et al., 2020;McCann et al., 1998). Expanding on Haeckel's definition, ecology now appears as the study of strong and weak interactions between the organism and the environment, and identifying strong interactions is one of its main goals.
A great help in solving this problem is that the frequency distribution of interaction strengths is not uniform; in fact, it is highly skewed such that there are few strong and many weak interactions, as first shown by Paine (1992) for a rocky intertidal community and corroborated by subsequent works (e.g., Jordán et al., 2003, Preston et al., 2018reviewed in Landi et al., 2018, Power et al., 1996, Wootton & Emmerson, 2005. Remarkably (and perhaps surprisingly), the idea of "few strong -many weak interactions" may even be a quarter of a century older than Haeckel's (1866) definition of ecology. It can be considered to go back to Liebig's law of the minimum, according to which there is only one limiting factor, for example, a nutrient in shortest supply (Liebig, 1840, see also Nikol'skii, 2014Berryman, 1993Berryman, , 1999Berryman, , 2003 provides a recent discussion of the law. The Liebig law can be viewed as the "few strong -many weak interactions" concept pushed to its limit: There is only one strong interaction, or equivalently a single limiting factor, and the rest are all weak. And vice versa, the concept can be regarded as a relaxed version of Liebig's law: As suggested by Berryman (1993), when the system is out of equilibrium, there may be not a single but a few strong interactions. It is this "few strong -many weak interactions" property that makes the task of identifying strong interactions feasible, for it is possible (or at least easier) to find one or a few strong effects but hardly so when there are many. Unfortunately, Liebig's law in both its strict (a single limiting factor) and relaxed (few strong-many weak interactions) forms acts as an existence theorem in mathematics: It posits the existence of a single or just a few strong interactions but says little about how to find them.
Contrary to expectations based on Liebig's law, a single strong effect often remains elusive. One of the reasons may be that temporal dynamics is a characteristic feature of ecological systems (e.g., Kerimoglu et al., 2013;reviewed in Leroux & Loreau, 2015), so individual strong interactions may operate over relatively short time intervals, and ability to identify them would depend in a major way on the temporal resolution of the methods used. To be more specific, in this study we focus on trophic interactions, namely bottom-up (food availability) and top-down (predation pressure) effects, in populations of zooplankton. These interactions play a pivotal role in the functioning of populations and communities in general (Leroux & Loreau, 2015) and in zooplankton in particular (Feniova et al., 2019;Gliwicz, 2003;Hampton et al., 2006;Huber & Gaedke, 2006;Liu et al., 2020;Marino et al., 2020). The key thing is that they tend to alternate over time. Thus, according to the seminal PEG (Plankton Ecology Group) model of seasonal succession in plankton communities (Sommer et al., 1986(Sommer et al., , 2012Straile, 2015), in temperate waters a spring increase in herbivorous zooplankton is typically caused by an abundant food supply while a subsequent decline is affected by fish predation as well. These observations are in good agreement with the general theory of cascading top-down effects (Leroux & Loreau, 2015;Oksanen et al., 1981) according to which food limitation and predation pressure would replace each other over time as major driving forces of population dynamics. Then, temporal averaging, which is a direct consequence of the insufficient temporal resolution of the methods, would result in bottom-up and top-down effects (or, for that matter, any factors) contributing roughly equally to population changes, giving a misleading and discouraging impression that "everything depends on everything else" (Berryman, 1993; in popular literature this formulation is known as Barry Commoner's first law). This problem is known by many names such as the "average temperature across the hospital" (Vilenkin, 1978) and, on a less ironic note, the fallacy of averaging (cf. Welsh et al., 1988), further emphasizing its importance. Our proposition is that the chance to detect a single governing factor or at least one clearly prevailing over the others would be higher if the method at hand has sufficient temporal resolution. Therefore, studying temporal resolution may be key to identifying strong interactions and thus to understanding population dynamics.
In this paper, we have studied the temporal resolution of one particular method, namely, contribution analysis of birth rate, which is intended to assess the relative strength of bottom-up versus top-down effects in cladoceran zooplankton (Polishchuk, 1995;Polishchuk et al., 2013). Birth rate is estimated by the Edmondson-Paloheimo model (Edmondson, 1968;Paloheimo, 1974). The model gives per capita birth rate as a function of fecundity (number of eggs per adult female), proportion of adults, and egg development rate. Fecundity and proportion of adults are taken as intermediaries between, respectively, food and predators, on the one hand, and birth rate, on the other. The effect of food and predators on birth rate and, by implication, on population dynamics is expressed as the product of the partial derivatives of birth rate with respect to fecundity and proportion of adults, respectively, times actual changes in fecundity and proportion of adults. The meaning of this calculation is that the partial derivatives yield a potential change in birth rate in response to a unit change in fecundity or proportion of adults, while the product of this potential change by the actual changes in fecundity and proportion of adults gives the component changes in birth rate caused by those population characteristics. After Caswell (1989Caswell ( , 2001: Ch. 10), who proposed this as a general approach, the products are termed contributions (of changes in fecundity and proportion of adults to the resulting change in birth rate), hence the name of the method, contribution analysis of birth rate (see Section 2 for further details). The next step in developing contribution analysis of birth rate has been to construct the ratio of contributions of changes in the proportion of adults and fecundity to change in birth rate; we call it the ratio of contributions, R (Polishchuk et al., 2013). The contributions are taken in absolute value, so R ≥ 0. The behavior of R under different combinations of top-down and bottom-up effects has been studied experimentally (Polishchuk et al., 2013). The laboratory and computer experiments on Daphnia have shown that R < 1 is indicative of strong bottom-up effects as compared to top-down effects. These results were obtained by considering 6 or 11 sampling intervals in laboratory and computer experiments, respectively. No attempt was made to assess whether the instructive values of the ratio of contributions could be obtained on fewer intervals, a question that is directly related to the temporal resolution of the method.
Thus, contribution analysis of birth rate has the potential to identify strong bottom-up effects, but neither have the above experimental findings been tested in the field nor has the method's temporal resolution been examined.
In this study, we are dealing with a herbivorous zooplankton species, the cladoceran Bosmina longirostris, inhabiting a small northern lake with few planktivorous predators and thus likely controlled by food. We describe this situation as one in which strong bottom-up effects are at place, based on the general theory of cascading topdown effects (Leroux & Loreau, 2015;Oksanen et al., 1981) according to which bottom-up and top-down effects are inextricably linked and tend to negatively correlate with one another (see above). Since predation effects on our Bosmina population are weak, bottom-up effects are to be strong (except for the initial phase of population growth, see Section 3). Note that in the previous experimental work (Polishchuk et al., 2013) to which we compare the results of the present study, strong bottom-up effects were defined in the same way: They were considered strong in microcosms (aquaria) where there was no predation other than sampling.
Using this Bosmina population as a test case, we pursue two goals.
First, we test whether contribution analysis of birth rate allows one to detect strong bottom-up effects in natural zooplankton populations. More specifically, we test the prediction, based on the independent experimental findings (Polishchuk et al., 2013), that under strong bottom-up effects the ratio of contributions R is less than 1.
Second, and most important, we determine the temporal resolution of contribution analysis of birth rate; that is, the minimum number of sampling intervals and the minimum length of time needed to reliably determine R and thus to identify strong bottom-up effects. Although the problem of temporal resolution is addressed here through one particular example, we believe it is a general one. Given that ecological systems typically vary over time and can move rather quickly from the state of being controlled by one factor to the state of being controlled by another, success in identifying strong interactions may very much depend on the temporal resolution of the method used.

| Study site and ecological environment
Data were collected at Vodoprovodnoye Lake, which is located near the Arctic Circle, in the vicinity of the White Sea Biological Station of Lomonosov Moscow State University. It is a small shallow freshwater lake with a maximum depth of about 2.7 m and a surface area of about 0.6 ha (Bizina, 2000;Mardashova et al., 2020; A. N. Pantyulin, personal communication). Secchi disk transparency regularly recorded during the study period in 2013 was 1.4 ± 0.2 m (mean ± SD; n = 23), which suggests a eutrophic state (Bigham Stephens et al., 2015). The lake is surrounded by forest and in some places by floating mats of peat mosses, and the water has a yellowish tint, likely due to the presence of humic substances. It is known that in colored shallow lakes such as this one Secchi disk depth may not be indicative of trophic state (Carlson, 2007).
On two occasions in 2013, we measured nutrient concentrations in the lake ( Table 1). According to the classification by Oksiyuk and Zhukinsky (Oksiyuk et al., 1993), which is popular in Russia and neighboring countries, the levels of phosphate and nitrate suggest that the lake is oligotrophic, while the levels of nitrite and ammonium correspond to a mesotrophic or a eutrophic state, respectively. Based on  Accessed on 20 July 2021) that stipulate that phosphate concentrations of less than 0.05 mg P L −1 indicate an oligotrophic state of the water body. Although in our case both phosphate and nitrate implicate an oligotrophic state, the latter has a readily available source of replenishment (ammonium) while the former does not; hence, the lake is likely to be P-limited. Finally, concentrations of algal pico-and nanoplankton observed in Vodoprovodnoye Lake during both study seasons, 2012 and 2013 (see Figure 1), were comparable to those reported for oligotrophic lakes (Stockner & Shortreed, 1989).
An interesting property of the vertical temperature distribution in this shallow lake (though probably not uncommon for small forest In 2012 and 2013 when we conducted our research, the only mesozooplankton species found in the pelagial of the lake were the cladoceran Bosmina longirostris, which is the focus of this study, and the rotifer Asplanchna priodonta (cf. Bizina, 2000). Given that Asplanchna is a predator (on small rotifers) and occasionally a grazer but in the latter case it feeds on large algae that cannot be ingested by Bosmina (Kappes et al., 2000), Bosmina had no species to compete with and thus may have experienced only intraspecific competition.
Among invertebrate predators that could potentially affect the cladoceran abundance, only Chaoborus larvae were present in the lake. However, there were so few of them, just about a dozen caught in the net during the entire two-season study period, that they seem unlikely to have had any significant effect on the Bosmina population. Note that Chaoborus were found in the gut of the ninespine stickleback (see below), but this does not necessarily show that they were abundant in the lake; rather, this may indicate that the fish fed selectively on them.
The only fish predator present in the lake was the ninespine stickleback (Pungitius pungitius). The stickleback invaded Vodoprovodnoye Lake after 1995, when it was temporarily connected to a nearby Verkhneye Lake by a ditch (Bizina, 2000). To estimate its pressure on the Bosmina population, in 2013 we used minnow traps baited with "TetraCichlid" fish food (Tetra, USA). The traps were placed along the shore on two occasions, three traps from 29 June to 3 July and 10 traps on 17 July for 24 h, the latter case corresponding to high Bosmina abundance (see Figure 1 where 17 July is Day 24). Overall, just four stickleback individuals were caughttwo on 3 July and two on 17 July, with body length and wet weight (mean ± SE) of 42.2 ± 1.3 mm and 0.90 ± 0.15 g. The analysis of fish gut contents showed that the ration predominantly consisted of insect remains, including Chaoborus larvae, and cladocerans, mainly Chydoridae. Only one specimen among several tens of items found resembled Bosmina. Based on this, we concluded that the Bosmina population did not experience any substantial predation pressure in Vodoprovodnoye Lake. Moreover, previously published data on zooplankton community dynamics in this lake (Bizina, 2000) suggest that even in 1997, when larger cladocerans were significantly affected by the then-recent invasion by the stickleback, Bosmina largely escaped its impact. On every sampling occasion, we measured water temperature at three depths of 0.1, 0.5, and 1 m from the water surface. Given that below 1 m Bosmina were very few (see above), to assess their egg development time, which is temperature-dependent (Bottrell et al., 1976), we used the average temperature at the abovementioned depths.

| Data collection
To count animals and eggs, a 5-to 20-mL subsample of the total sample, depending on the abundance of zooplankton, was placed in a Bogorov tray, fixed with several drops of 4% buffered formalin and counted at 32× with a stereomicroscope. Only female

| Calculating birth rate
The original Edmondson-Paloheimo model (Edmondson, 1968, his equation 13;Paloheimo, 1974) to calculate per capita birth rate b in zooplankton, in particular in cladocerans, is To make the model better suitable for contribution analysis, it can be equivalently rewritten in terms of egg development rate V, fecundity F = E/N a and proportion of adults A = N a /N (Polishchuk, 1995).
The Edmondson-Paloheimo model is based on rather general assumptions (Polishchuk et al., 2013;Voronov, 1991) and thus describes birth rate quite well (Mooij et al., 2003, and references therein). Written in form (1), it allows one to relate birth rate to the quantity and quality of food (through F), predation pressure (through A; because in zooplankton predators are typically size-selective), and temperature (through V; Polishchuk, 1995;Polishchuk et al., 2013).
Thus, the Edmondson-Paloheimo model written in form (1) helps to build the bridge from the major environmental factors (food, or bottom-up effect; predation, or top-down effect; and temperature) through the mediation of the population characteristics (F, A, V, respectively) to birth rate.

| Calculating contributions
Contribution analysis (Caswell, 1989) as applied to the Edmondson-Paloheimo birth rate model is a method to quantify the effect of changes in F, A, and V on the resulting change in b and thus to express the effects of the environmental factors that underlie F, A, and V on population dynamics, using birth rate (or to be more precise, change in birth rate) as a common currency (Polishchuk, 1995, Polishchuk et al., 2013. The basic equation of contribution analysis of birth rate is where the first term on the right-hand side is the contribution of a change in fecundity ΔF to the resulting change in birth rate Δb and similarly for the second and third terms associated with the contributions of ΔA and ΔV. The sum on the right-hand side, however, equals Δb only if shifts in F, A, and V are all sufficiently small. Often this will not be the case because those shifts represent actual population changes between successive sampling dates. To overcome this problem, we rewrite Equation (2) in terms of the derivatives with respect to time t and integrate the left-hand side and, term by term, the right-hand side of Equation (3) It may be of interest to note that the above approach is a general one: it does not depend on the form of a particular function to be decomposed into contributions. It is always possible to replace instantaneous partial derivatives with the average values, thus making the sum of contributions equal to the resulting change in the function of interest.
Quantitatively, contributions from Equations (4)-(6) were obtained through numerical integration using a computer code written in QBasic (Appendix S1; the credit for the invention of the integral form of the contributions belongs to D. A. Voronov, who also wrote the QBasic code; for more details about the integration see Polishchuk et al., 2013).

| Calculating the ratio of contributions
In order to detect the presence, and perhaps to measure the strength, of bottom-up effects in the lake, we employ the ratio of contributions, one of which, that from Equation (5), is associated with change in the proportion of adults and the other, that from Equation (4), with change in fecundity. Those contributions refer to a certain sampling interval. Because a single interval may not be sufficient to obtain reliable estimates of the ratio, it is calculated over a time period spanning multiple consecutive sampling intervals. We have used the following 4-step procedure to find the characteristic ratio of contributions over multiple sampling intervals (Polishchuk et al., 2013): (1) Contributions per sampling interval from Equations (4) and (5)  Hence, we cannot and do not evaluate the statistical significance of the difference in R associated with different number of intervals.
Instead, we compare our field estimates of R with the laboratory values obtained in the independent experiments (Polishchuk et al., 2013).

| RE SULTS
Although the Bosmina population did not experience any substantial predation pressure and hence top-down effects were weak (see Section 2.1), it does not necessarily follow that bottom-up effects were invariably strong. In the first part of the season when population sizes were relatively low, food may have been in excess, with no considerable bottom-up effects to occur. Hence, our first task was, focusing on the second part of the season, to identify the periods of food shortage when bottom-up effects were likely to be at work. These were the periods for which we calculated the ratio of contributions and compared it with the previously obtained experimental results.
Given little predation pressure, periods of food shortage are those where negative density dependence manifests itself (Hixon & Johnson, 2009). The negative density dependence is defined as a negative relationship between per capita population growth rate year from interval of length 3 on, all R-values lie below the lower experimental error bound (except for the value pertinent to 11 sampling intervals, which is excluded from the calculation of the averages above, see Figure 4). Thus, in both years R curves follow a declining plateauing pattern, and the plateau is lower in 2013 than in 2012.

| DISCUSS ION
In this study, we tested in the field a method of analysis of birth rate dynamics in zooplankton, termed contribution analysis of birth rate, and its derived metric, the ratio of contributions, R. To be more specific, we tested in the field whether R is indicative of strong bottom-up effects when top-down effects are weak or virtually absent.
Furthermore-and this is perhaps even more important-we determined the temporal resolution of the method. The previous laboratory and computer experiments (Polishchuk et al., 2013) suggested that the R ratio might be useful to assess the relative strength of topdown versus bottom-up effects. The lake we worked on is a simple ecosystem with few planktivorous predators; that is, only bottom-up effects are likely to be at work there. The system is further simplified by there being only one common species of crustacean zooplankton in the pelagial, the cladoceran Bosmina longirostris, which F I G U R E 3 Birth rate analysis in Bosmina longirostris in 2012 and 2013. The analysis consists of decomposing changes in per capita birth rate b, Δb, into three contributions, each due to changes in only one population characteristic, namely egg development rate, ConV, fecundity, ConF, or proportion of adults, ConA, and then calculating the ratio of contributions, |ConA| over |ConF| (see Figure 4). Each cluster of bars represents the decomposition of Δb for a given sampling interval. In every cluster, the left-most bar indicates Δb divided by the length of the interval, T, in days; and the other three bars are contributions expressed on a per day basis. The contributions add up to the change in birth rate, that is, (ConV) day + (ConF ) day + (ConA) day = Δb/T. Note that bars above or below the zero line reflect an increase or decrease in the corresponding characteristic in a given sampling interval. Rectangles delineate the periods in the second part of the season where density dependence manifests itself most clearly, and thus, bottom-up effects are likely to be most pronounced. The ratio of contributions shown in Figure 4 refers to just these periods. Thus, contribution analysis of birth rate, and the ratio of contributions based on it, has successfully passed the field test and, most probably, can be used to identify strong bottom-up effects in natural zooplankton populations. Furthermore, the temporal resolution of the method is found to be as short as 2-4 sampling intervals, or 6-12 days.
The temporal resolution of contribution analysis of birth rate revealed in this study has a clear biological meaning. In Table 2 Polishchuk et al., 2013). Common to the field and laboratory populations is the key role of food in population dynamics; hence, the general agreement between field and laboratory R-values, seen in the graph starting from two sampling intervals, is evidence that R is an indicator of trophic conditions in the field. The graph also shows that two to four sampling intervals ensure minimum averaging needed to obtain the ratio of contributions characteristic of a given year. This indicates the temporal resolution of the method. The ratio of contributions in the field refers to 6 and 11 last sampling intervals in 2012 and 2013, respectively, a period where bottom-up effects are likely to be most pronounced. The ratio of contributions observed in the laboratory is provided with 1 SE error bounds, R-exp bounds; note that the upper and lower bounds are not exactly symmetrical about the mean because they are back-calculated from the logtransformed R. is found to be 3.7-9.2 days ( Table 2, all rows except the last one; temperature for 2013, and not for 2012, is taken because detailed body-size measurements, to be used below, are available only for that year). We also calculated D m directly for our Bosmina population using regression equations that relate D m to water temperature and body size in cladocerans (Gillooly, 2000;Gillooly et al., 2002); this gives 5.3-5.7 days ( Table 2,  The above result has an interesting and important consequence: The temporal resolution of the method seems comparable to the lifetime of a single strong interaction. There are two arguments and a guess to support this assertion. The first argument is based on the simple observation made by planktonologists that in temperate lakes in the course of the season zooplankton populations often go through two or three population maxima and minima, so each rise or fall would last one or two or perhaps 3 weeks. The second argument, based on modeling (Higgins et al., 1997;Ricker, 1954), is that the period of population oscillations tends to be twice the generation time; hence, one generation time approximates the duration of a population increase or decline. Given that the generation time in cladocerans, a major component of zooplankton, is about one or two weeks (see, e.g., Table 2 for the time to maturity in Bosmina, but note that it is shorter than the generation time), and may be even longer in the case of food shortages (Romanovsky, 1984), the second argument leads to approximately the same duration of monotonic population changes as does the first. Our guess is that, often, there may be only one major driving force, for example, food or predators, responsible for a single population growth or decline event (cf. driver-response relationships in time series, Ryo et al., 2019). This reasoning suggests that contribution analysis of birth rate has sufficient temporal resolution to detect the effect of food on major population-dynamics events just "on the fly," right at the time that the effect occurs.
Contrary to the expectation that the length of monotonic segments of population change is about the generation time, our Bosmina TA B L E 2 Time to maturity of Bosmina and additional relevant information.

Experimental data (except for our data in the last row) Calculated
Body length at maturity, mm Bosmina longispina maritima d 0.39 ± 0.02 (±SD) 1.3 ± 0.5 (±SD) 10 10.6 ± 1.0 (±SD) 5.0-6.1 Bosmina longispina maritima d NA 2.8 ± 0.8 (±SD) 15 5.9 ± 0.2 (±SD) 4.6-4.9 Bosmina longirostris e 0.27 ± 0.01 (±SE) 1.1 ± 0.1 (±SE) 20 5.8 ± 0.8 (±SE) 6.1-8.0 Our data, 2013 0.25 ± 0.02 (±SD) 1.0 ± 0.2 (±SD) 17.6 NA 5.3-5.7 Note: D m1 , time to maturity measured in the experiments (literature data); D m2 , time to maturity adjusted to 17.6°C, which is the average lake temperature in 2013. For the experimental data, D m2 is calculated on the basis of D m1 using a temperature correction from the Arrhenius equation: D m2 = D m1 exp(6962.98(1/(17.6 + 273) -1/(T c1 + 273))) where T c1 is temperature in the experiments in degrees Celsius, and 6962.98 is the ratio of the activation energy (0.6 eV) to Boltzmann's constant (8.617 × 10 −5 eV K −1 ) expressed in kelvin (K) (based on Gillooly et al., 2002; all rows except the last one); range of D m2 is calculated from the range of D m1 which is shown either directly or as mean ± SD or mean ± SE, depending on what (if any) is provided in the original publications. For our Bosmina population, D m2 is calculated from the water temperature and body mass, either on the basis of the equation given in Gillooly et al. (2002: figure 4; first value in the last row of D m2 ) or on the basis of the equation for postembryonic development time (Gillooly, 2000; second value in the last row of D m2 ). Body mass is calculated as M = 26.6 L 3.13 where M is dry body mass in μg, L is body length in mm (Dumont et al., 1975). Body length and clutch size presented in the table serve to generally characterize animals and trophic conditions in the experiments and in our lake. Our data refer to 2013 only because in 2012 Bosmina were not accurately measured for body length.
e Urabe (1991) (data at food concentration of 0.05 mg C L −1 , which most closely corresponds to the characteristics of Bosmina in the present study).
populations continuously grew several times longer ( Figure 1).
Apparently, from late June to early August in both years of study, the lake's system was in the state of a prolonged biological spring, which is evidenced by the similarity between the population growth curves in this northern lake and those observed during the calendar spring in temperate lakes as described by the PEG model (Sommer et al., 1986(Sommer et al., , 2012. Given that contribution analysis of birth rate allows one to detect the main factor of population dynamics on relatively short intervals of about the generation time, it will be all the more able to do so on longer intervals. This simply follows from the fact that on intervals longer than the generation time the R curves generally plateau-and just at the expected values (see Figure 4).
Our The results of this study are obtained with the use of the Edmondson-Paloheimo model (Edmondson, 1968;Paloheimo, 1974).
It has been repeatedly shown that the model relies on rather general assumptions and approximates quite well birth rate in zooplankton, especially in cladocerans (e.g., Hairston, 2015;Mooij et al., 2003;Polishchuk, 1986). There are a number of mathematical derivations of the model (the most elegant and general one is due to Voronov, 1991), which all link birth rate to time intervals equal to D, the egg development time, thus implying that an interval of length D is an adequate sampling interval to estimate birth rate (Polishchuk, 1982(Polishchuk, , 1986. Given that the egg development time at normal summer temperatures is about 3 or 4 days, this is the reason why in this study sampling intervals are chosen to be just that length. There are two general methods commonly used to assess the strength of ecological interactions: One is based on matrices of partial derivatives (Jacobian matrices;de Ruiter et al., 1995;Novak et al., 2016) and the other makes use of multivariate autoregressive (MAR) models (Gabaldón et al., 2019;Hampton et al., 2006;Huber & Gaedke, 2006;Ives, 1995;Ives et al., 1999). Contribution analysis of birth rate involves partial derivatives and thus belongs to the former type. Table 3 provides a detailed comparison between our approach and one form of the Jacobian matrices, a community matrix, which is a matrix of partial derivatives of the population growth rates of each species with respect to population size of every species in the community (de Ruiter et al., 1995;Novak et al., 2016). The major difference is that our approach takes into account not only partial derivatives but also actual changes in the quantities of interest between sampling occasions, namely Δ(fecundity) and Δ(proportion of adults), which makes it more dynamically oriented. Thinking in terms of dynamics when "everything changes and nothing stands still" (Heraclitus) readily raises the question of the method's temporal resolution (does the method allow us to capture those changes when they occur?), which is addressed in this study. This same question can be asked for other methods but, to the best of our knowledge, it is often overlooked. Furthermore, we suggest that the temporal resolution of about the lifetime of a single strong interaction, which we came across in relation to one particular method, contribution analysis of birth rate, in a particular context, bottom-up and top-down effects in zooplankton, would be a generally desirable property for any method intended to identify and assess any type of strong ecological interactions, for it enables one to keep up with interaction changes.
We hope that our study will not only draw attention to the problem of temporal resolution but will also stimulate the search for methods possessing the above property.
It is a popular belief that Nature is extremely complex, or, as some people would put it, "everything depends on everything else." This discouraging picture may be misleading, though. One of the reasons for this misconception is that the methods used often do not have sufficient temporal resolution or the problem of resolution is simply ignored. The lack of resolution will automatically lead to averaging over long time intervals, resulting in multiple factors contributing roughly equally to the process of interest and thus indeed giving the impression, a false one, that everything depends on everything else. The situation may be different on short time intervals, where just a single strong interaction is likely to occur. But how short do those short intervals need to be? The answer "the shorter the better" seems neither practical nor heuristically useful. Let's put it differently: What is a time interval, not too short but a reasonable one, sufficient to identify strong interactions? The answer proposed in this paper is that, conceptually, its length equals the lifetime of a single strong interaction.
Furthermore, in the context of population dynamics, the interval can be defined more precisely: It is about the generation time of the species of interest. This is because the periods of population increase or decline are often comparable to the species' generation time, and each of those increase or decline events may be associated with a single major driving force or, equivalently, a single strong interaction. We came up with these ideas inductively TA B L E 3 Comparative table to show similarities and differences between the methods of assessment of the strength of top-down and bottom-up effects using community matrices (left section) and using contribution analysis of birth rate (right section). Step

Interaction strength in a community matrix
Interaction strength by means of contribution analysis of birth rate Decomposition of a change in Ẋ 1 or b into components, each due to a change in the variable involveḋ A working model for population growth rate per capita birth rate in zooplanktoṅ Self-effect (not of interest here) Temperature effect (not of interest here) Note: The methods are divided into five (community matrices) or six (contribution analysis) steps, as indicated. The main similarity between the methods is that both employ partial derivatives to measure interaction strengths, and the main difference is that contribution analysis, after Caswell (1989), additionally accounts for actual changes in the variables considered so that the strength of the effect is expressed as the partial derivative with respect to a given variable times the actual shift in that variable (Step 5). An additional difference is that contribution analysis of birth rate involves population characteristics (fecundity and proportion of adults) that are intermediate between the environmental factors (food and predation, respectively) and the population's response (Step 1a; this step is not available, NA, in the community matrices method). Community matrices section. X 1 , X 2 , and X 3 are the abundances of a target species, food, and predators, respectively. Arrows indicate the effect, including a selfeffect (Step 1). In the left-hand side of the expression for decomposition (Step 3), there is Ẋ 1 rather than ΔẊ 1 because the expansion is performed around the equilibrium point at which Ẋ 1 = 0. The model shown at Step 4 is a generalized Lotka-Volterra model (Yodzis, 1989) commonly used in the context of community matrices (Novak et al., 2016). Contribution analysis of birth rate section. Arrows indicate the effect (Steps 1 and 1a). Predators are assumed to be size-selective, as is often the case in zooplankton, hence the effect of predators on the proportion of adults A and through A on per capita birth rate b (Step 1a). The model shown at Step 4 and used in this study and elsewhere (Polishchuk, 1995;Polishchuk et al., 2013) is the Edmondson-Paloheimo model of birth rate in zooplankton (Edmondson, 1968;Paloheimo, 1974). Along with fecundity and proportion of adults, the model includes the egg development rate V, which is the reciprocal of the egg development time. The latter is known to largely depend on temperature; its effect on birth rate, and hence the effect of temperature, is not considered here. More information is given in the body of the table.
while exploring one particular method, contribution analysis of birth rate in zooplankton, but believe they are of general interest.
The method's temporal resolution is found to be approximately the generation time of the species to which it has been applied.
Importantly, the method does not allow us to get rid of averaging altogether; some averaging is necessary, since an interval of the length of generation time must accommodate two to four sampling intervals over which the averaging is performed. Given our species' generation time, this means that each sampling interval is to be 3-4 days, which is the case in this study. It remains to be seen whether, and to what extent, averaging is a general requirement, that is, how the necessity for temporal resolution interacts with the need for temporal averaging. To conclude, our work shows that contribution analysis of birth rate is a promising tool to identify strong trophic interactions in zooplankton. In a broader perspective, this study suggests that dynamically oriented methods with a temporal resolution of about the lifetime of a single strong interaction may be key in disentangling strong interactions in ecological communities and thus in solving one of the main issues in ecology.

CO N FLI C T O F I NTER E S T S TATEM ENT
No conflicts of interest.

DATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are openly available in Figshare at https://doi.org/10.6084/m9.figsh are.21268512.
A QBasic computer code to calculate contributions, file "ConBasic.
BAS," is available in Appendix S1. Two more supplementary files are "Datain.txt," which is an example of an input data file, and "CONTS.
txt," which contains the output data, that is, contributions, calculated from the input data. The input file shows how input data must be arranged, and the output file can be used to check whether the computer code works correctly. For further analysis of the contribution results, the CONTS.txt file can be converted into an Excel spreadsheet. The fourth supplementary file, "Description_of_files_ used_to_calculate_contributions," as the name suggests, details the content of the above three files.